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				<span dir="auto">Digital biquad filter</span>
			</h1>
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				<div id="mw-content-text" dir="ltr" class="mw-content-ltr" lang="en"><p>In <a href="http://en.wikipedia.org/wiki/Signal_processing" title="Signal processing">signal processing</a>, a <b>digital biquad filter</b> is a second-order <a href="http://en.wikipedia.org/wiki/Recursive_filter" title="Recursive filter">recursive</a> <a href="http://en.wikipedia.org/wiki/Linear_filter" title="Linear filter">linear filter</a>, containing two <a href="http://en.wikipedia.org/wiki/Pole-zero_diagram" title="Pole-zero diagram" class="mw-redirect">poles</a> and two <a href="http://en.wikipedia.org/wiki/Pole-zero_diagram" title="Pole-zero diagram" class="mw-redirect">zeros</a>. "Biquad" is an abbreviation of "<i>biquadratic</i>", which refers to the fact that in the <a href="http://en.wikipedia.org/wiki/Z-transform" title="Z-transform">Z domain</a>, its <a href="http://en.wikipedia.org/wiki/Transfer_function" title="Transfer function">transfer function</a> is the ratio of two <a href="http://en.wikipedia.org/wiki/Quadratic_function" title="Quadratic function">quadratic functions</a>:</p>
<dl>
<dd><img class="tex" alt="\ H(z)=\frac{b_0+b_1z^{-1}+b_2z^{-2}} {1+a_1z^{-1}+a_2z^{-2} }" src="Digital_biquad_filter_pliki/ec8f24be333ba0478587a2e8d2bff50b.png"></dd>
</dl>
<p>High-order recursive filters can be highly sensitive to <a href="http://en.wikipedia.org/wiki/Quantization_%28signal_processing%29" title="Quantization (signal processing)">quantization</a> of their coefficients, and can easily become <a href="http://en.wikipedia.org/wiki/BIBO_stability" title="BIBO stability">unstable</a>.
 This is much less of a problem with first and second-order filters; 
therefore, higher-order filters are typically implemented as 
serially-cascaded biquad sections (and a first-order filter if 
necessary).</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
<span class="toctoggle">&nbsp;[<a href="#" class="internal" id="togglelink">hide</a>]&nbsp;</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Implementation"><span class="tocnumber">1</span> <span class="toctext">Implementation</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Direct_Form_1"><span class="tocnumber">1.1</span> <span class="toctext">Direct Form 1</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="#Direct_Form_2"><span class="tocnumber">1.2</span> <span class="toctext">Direct Form 2</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-4"><a href="#References"><span class="tocnumber">2</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#See_also"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Digital_biquad_filter&amp;action=edit&amp;section=1" title="Edit section: Implementation">edit</a>]</span> <span class="mw-headline" id="Implementation">Implementation</span></h2>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Digital_biquad_filter&amp;action=edit&amp;section=2" title="Edit section: Direct Form 1">edit</a>]</span> <span class="mw-headline" id="Direct_Form_1">Direct Form 1</span></h3>
<p>The most straightforward implementation is the Direct Form 1, which has the following <a href="http://en.wikipedia.org/wiki/Difference_equation" title="Difference equation" class="mw-redirect">difference equation</a>:</p>
<dl>
<dd><img class="tex" alt="\ y(n) = b_0x(n) + b_1x(n-1) + b_2x(n-2) - a_1y(n-1) - a_2y(n-2) " src="Digital_biquad_filter_pliki/3c13155d7d9c27c20daec2a1f139f9b4.png"></dd>
</dl>
<p>Here the <img class="tex" alt="b_0" src="Digital_biquad_filter_pliki/3e5dc8a9e58fac43ec3377c25606be6b.png">, <img class="tex" alt="b_1" src="Digital_biquad_filter_pliki/baacbe90a611155ee8de5fa5e833e134.png"> and <img class="tex" alt="b_2" src="Digital_biquad_filter_pliki/56ece28df930bd4d153b6c74ba96c054.png"> coefficients determine zeros, and <img class="tex" alt="a_1" src="Digital_biquad_filter_pliki/8e6ba967645c302e1f2a60ec9c341e5c.png">, <img class="tex" alt="a_2" src="Digital_biquad_filter_pliki/0f768ac5d5dea8d93716a27da05871de.png"> determine the position of the poles.</p>
<p>Flow graph of biquad filter in Direct Form 1:</p>
<p><a href="http://en.wikipedia.org/wiki/File:Biquad_filter_DF-I.svg" class="image"><img alt="Biquad filter DF-I.svg" src="Digital_biquad_filter_pliki/400px-Biquad_filter_DF-I.png" width="400" height="208"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Digital_biquad_filter&amp;action=edit&amp;section=3" title="Edit section: Direct Form 2">edit</a>]</span> <span class="mw-headline" id="Direct_Form_2">Direct Form 2</span></h3>
<p>The Direct Form 1 implementation requires four delay registers. An 
equivalent circuit is the Direct Form 2 implementation, which requires 
only two delay registers:</p>
<p><a href="http://en.wikipedia.org/wiki/File:Biquad_filter_DF-II.svg" class="image"><img alt="Biquad filter DF-II.svg" src="Digital_biquad_filter_pliki/400px-Biquad_filter_DF-II.png" width="400" height="208"></a></p>
<p>The Direct Form 2 implementation is called the canonical form, 
because it uses the minimal amount of delays, adders and multipliers, 
yielding in the same transfer function as the Direct Form 1 
implementation. The <a href="http://en.wikipedia.org/wiki/Difference_equation" title="Difference equation" class="mw-redirect">difference equations</a> for DF2 are:</p>
<dl>
<dd><img class="tex" alt="\ y(n)=b_0 w(n)+b_1 w(n-1)+b_2 w(n-2)," src="Digital_biquad_filter_pliki/6695baf58da9f3a66640528797eb0f4a.png"></dd>
</dl>
<p>where</p>
<dl>
<dd><img class="tex" alt="\ w(n)=x(n)-a_1 w(n-1)-a_2 w(n-2)." src="Digital_biquad_filter_pliki/f82c9d8288c3b4765ada99d0a473bd85.png"></dd>
</dl>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Digital_biquad_filter&amp;action=edit&amp;section=4" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<ul>
<li><a rel="nofollow" class="external text" href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470014954.html">Sen M Kuo, Bob H Lee: <i>Real-Time Digital Signal Processing</i>, Wiley &amp; Sons Ltd, 2001</a></li>
<li><a rel="nofollow" class="external text" href="http://www.musicdsp.org/files/Audio-EQ-Cookbook.txt">Cookbook formulae for audio EQ biquad filter coefficients</a></li>
<li><a rel="nofollow" class="external text" href="http://peabody.sapp.org/class/350.838/lab/biquad/">Biquad filter</a></li>
<li><a rel="nofollow" class="external text" href="http://ccrma.stanford.edu/%7Ejos/filters/BiQuad_Section.html">JOS BiQuad section</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Digital_biquad_filter&amp;action=edit&amp;section=5" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Biquad_filter" title="Biquad filter" class="mw-redirect">Biquad filter</a></li>
<li><a href="http://en.wikipedia.org/wiki/Digital_filter" title="Digital filter">Digital filter</a></li>
</ul>


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